📐 geometry

1. **Nêu bài toán:** Cho tam giác GEM vuông tại M, với $EM = \frac{1}{2} EG$. Tìm các góc $\angle MEG$ và $\angle EGM$. 2. **Công thức và quy tắc:** Trong tam giác vuông, theo định

1. **Problem statement:** Given an isosceles triangle $BCD$ with vertex at $B$, points $E$ on $BC$ and $F$ on the ray opposite to $DB$ such that $CE = DF$. Lines $EH$ and $FG$ are

1. The problem asks to compile a list of angle properties and characteristics, focusing on symmetry and angle properties. 2. Angle properties include:

1. **State the problem:** Find the surface area of a rectangular pyramid with length $l=13$ cm, width $w=11$ cm, and height $h=8$ cm. 2. **Formula and explanation:** The surface ar

1. **State the problem:** Find the surface area of a rectangular pyramid with length $l=8$, width $w=4$, and height $h=2$. The surface area includes the base area plus the area of

1. **Problem Statement:** Find the length of $\overline{BL}$ given that $\overline{TS}$ is parallel to $\overline{AL}$, with $AL=12$, $TS=7.75$, and $SL=5.1$.

1. **Problem Statement:** Given a parallelogram PQRS with points T and M inside it such that $PT = MR$ and $PT \parallel MR$, prove a certain property (usually this type of problem

1. **State the problem:** We are given a sector OAB of a circle with radius $r=5.9$ mm and area $28$ mm$^2$. We need to find the central angle $\theta$ in degrees, rounded to 1 dec

1. The problem is to determine if line segments CD and EF are coplanar. 2. Two line segments are coplanar if they lie on the same plane.

1. **Problem Statement:** We have two index cards labeled with points A, B, C, D, E, F, G, M, N and lines AB, CD, EF, and planes M and N. We need to find intersections of lines and

1. **State the problem:** We have an isosceles triangle with two equal sides each measuring 11.52 inches and a vertex angle of 32° 15'. We need to find the length of the base.

1. **State the problem:** We have a circle with center G and points I, V, E, N on the circumference. Given arcs: - Arc VE = $3x - 5$

1. **Stating the problem:** We need to find the values of angles $x$, $y$, and $z$ given some angles and relationships. 2. **Analyzing the given information:** The problem shows an

1. **State the problem:** We have two similar triangles $ABC$ and $PQR$ with sides $AB=5.5$, $BC=5$, $CA=2.5$, and $PQ=16.5$, $QR=15$, $PR=n$. We need to find $n$. 2. **Recall the

1. **State the problem:** We have a solid hemisphere H with radius $x$ cm and volume $6174\pi$ cm³. A bowl is made by removing a smaller solid hemisphere from H, leaving a uniform

1. **Stating the problem:** We have three intersecting lines forming angles labeled $a\angle 5$, $44\angle 5$, $66\angle 5$, $c\angle 5$, and $b\angle 5$. We need to find the measu

1. **Stating the problem:** We are given a figure with angles labeled involving the Greek letter \(\xi\) and numbers 44, 66, and angles \(a\), \(b\), and \(c\) to find.

1. **State the problem:** We need to find the size of angle $\angle CME$ in the triangular prism $ABCDEF$ where $M$ is the midpoint of $AC$, $AC=40$ cm, $\angle CAE=35^\circ$, and

1. **State the problem:** We have a square with area 100 cm² inscribed inside a circle. We need to find the area of the circle. 2. **Recall formulas:**

1. **State the problem:** We need to find the area of a right-angled triangle with perpendicular sides of lengths 3 m and 14 m. 2. **Formula:** The area $A$ of a right-angled trian

1. The problem asks for the coordinates of point M, which is the midpoint of the line segment CD. 2. The midpoint formula for two points $C(x_1, y_1)$ and $D(x_2, y_2)$ is: